The np-completeness of the hamiltonian cycle problem in planar diagraphs with degree bound two

Plesník, Ján

doi:10.1016/0020-0190(79)90023-1

Cited by 97 publications

(67 citation statements)

References 5 publications

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“…Indeed, if deleting a set of edges from a cubic graph G results in an Eulerian graph G , then each vertex in G must have degree 2, so G must be a Hamiltonian cycle of G. Since the HAMILTONIAN CYCLE problem restricted to cubic planar graphs is NP-hard [16] the result follows. The directed version can be treated in a similar way using NP-hardness from [30].…”

Section: Directed Eulerian Edge Deletionmentioning

confidence: 99%

Parameterized Complexity of Eulerian Deletion Problems

Cygan

Marx

Pilipczuk

et al. 2011

Graph-Theoretic Concepts in Computer Science

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We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting contrast: while the node-deletion variants remain intractable, i.e., W[1]-hard for all the studied cases, edge-deletion problems are either fixed-parameter tractable or polynomial-time solvable. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the color coding technique. For versions that remain NP-complete but fixed-parameter tractable we consider also possibilities of

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Section: Directed Eulerian Edge Deletionmentioning

confidence: 99%

Parameterized Complexity of Eulerian Deletion Problems

Cygan

Marx

Pilipczuk

et al. 2011

Graph-Theoretic Concepts in Computer Science

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“…The 2DHCP consists of asserting whether or not D H contains a Hamiltonian circuit, that is, a closed walk traversing all the vertices of V H exactly once. The 2DHCP is known to be NP-complete [3]. We remark that the proof given by Plesnik [3] was devised for planar digraphs with indegrees and outdegrees at most two, yet by considering some additional arcs, we can easily extend this result to planar digraphs (with possible multiple arcs) with indegrees and outdegrees two.…”

Section: Np-completeness Of the Ecwppcpmentioning

confidence: 96%

On the complexity of the Eulerian closed walk with precedence path constraints problem

Kerivin

Lacroix

Mahjoub

2010

Electronic Notes in Discrete Mathematics

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The Eulerian closed walk problem in a digraph is a well-known polynomial-time solvable problem. In this paper, we show that if we impose the feasible solutions to fulfill some precedence constraints specified by paths of the digraph, then the problem becomes NP-complete. We also present a polynomial-time algorithm to solve this variant of the Eulerian closed walk problem when the paths are arcdisjoint. We also give necessary and sufficient conditions for the existence of feasible solutions in this polynomial-time solvable case.

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“…[15]. We modify Plesník's construction to create a gap-preserving reduction, then apply our reduction on Max-3SAT(29).…”

Section: Inapproximability Of Max Vertex-disjoint Path Cover(2)mentioning

confidence: 99%

“…This construction is largely the same as that from Ref. [15], but in- is set to True. Edge e 1 corresponds to a True literal (x i ), so it must not be traversed when we visit the clause side of the graph; edge e 2 corresponds to a False literal (x i ) and therefore must be traversed.…”

Section: Max-3sat(29) =⇒ Max Vertex-disjoint Path Cover(2)mentioning

confidence: 99%

Even 1 × <i>n</i> Edge-Matching and Jigsaw Puzzles are Really Hard

Bosboom

Demaine

et al. 2017

Journal of Information Processing

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Abstract:We prove the computational intractability of rotating and placing n square tiles into a 1 × n array such that adjacent tiles are compatible-either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999702 (On the other hand, there is an easy 1 2 -approximation). This is the first (correct) proof of inapproximability for edge-matching and jigsaw puzzles. Along the way, we prove NP-hardness of distinguishing, for a directed graph on n nodes, between having a Hamiltonian path (length n − 1) and having at most 0.999999284(n − 1) edges that form a vertex-disjoint union of paths. We use this gap hardness and gap-preserving reductions to establish similar gap hardness for 1 × n jigsaw and edge-matching puzzles.

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The np-completeness of the hamiltonian cycle problem in planar diagraphs with degree bound two

Cited by 97 publications

References 5 publications

Parameterized Complexity of Eulerian Deletion Problems

Parameterized Complexity of Eulerian Deletion Problems

On the complexity of the Eulerian closed walk with precedence path constraints problem

Even 1 × <i>n</i> Edge-Matching and Jigsaw Puzzles are Really Hard

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